Filtering of Markov jump processes by discretized observations

The article is devoted to a solution of the optimal filtering problem of a homogenous Markov jump process state. The available observations represent time increments of the integral transformations of the Markov state corrupted by Wiener processes. The noise intensity is also state-dependent. At the instant of the consecutive observation obtaining, the optimal estimate is calculated recursively as a function of previous estimate and the new observation, meanwhile between observations the filtering estimate is a simple forecast by virtue of the Kolmogorov differential system. The recursion is rather expensive because of need to calculate the integrals, which are the location-scale mixtures of Gaussians. The mixing distributions represent the occupation of the state in each of possible values during the mid-observation intervals. The paper contains numerically cheaper approximations, based on the restriction of the state transitions number between the observations. Both the local and global characteristics of approximation accuracy are obtained as functions of the dynamics parameters, mid-observation interval length, and upper bound of transitions number.